I will first state my question, and then give all the relevant definitions.

Q. Let $H$ and $K$ be monoids, and assume $H$ is essentially equimorphic to $K$. Is it true that $H$ is atomic if and only if so is $K$?

I can prove the "if" part, but I am baffled by the other direction.

We denote by $\mathscr{F}^\ast(\mathscr{U})$, for a fixed set $\mathscr{U}$, the free monoid with basis $\mathscr{U}$. We use the symbol $\ast$ for the operation of $\mathscr{F}^\ast(\mathscr{U})$, and we take $\|1_{\mathscr{F}^\ast(\mathscr{U})}\|_\mathscr{U} := 0$ and $\|z_1 * \cdots * z_n\|_\mathscr{U} := n$ for all $z_1, \ldots, z_n \in \mathscr{U}$. Given $\mathfrak z \in \mathscr{F}^\ast(\mathcal A(H))$, we call $\|\mathfrak z\|_\mathscr{U}$ the length of $\mathfrak z$.

With this in hand, let $H$ be a multiplicatively written monoid. We denote by $H^\times$ the set of units (or invertible elements) of $H$, by $\mathcal A(H)$ the set of atoms of $H$ (an element $a \in H$ is an atom if $a \notin H^\times$ and there do not exist $x, y \in H \setminus H^\times$ such that $a = xy$), by $\pi_H$ the unique homomorphism $\mathscr{F}^\ast(H) \to H$ such that $\pi_H(x) = x$ for every $x \in H$, and by $\mathscr{C}_H$ the smallest (monoid) congruence on $\mathscr{F}^\ast(\mathcal A(H))$ determined by the following condition:

• If $\mathfrak a = a_1 \ast \cdots \ast a_m$ and $\mathfrak b = b_1 \ast \cdots \ast b_n$ are, respectively, non-empty $\mathcal A(H)$-words of length $m$ and $n$, then $(\mathfrak a, \mathfrak b) \in \mathscr{C}_H$ if and only if $\pi_H(\mathfrak a) = \pi_H(\mathfrak b)$, $m = n$, and $a_1 \simeq_H b_{\sigma(1)}, \ldots, a_n \simeq_H b_{\sigma(n)}$ for some $\sigma \in \mathfrak S_n$.

Here, $\mathfrak S_n$ is the group of permutations of $[\![ 1, n ]\!]$, and $x \simeq_H y$, for $x, y \in H$, means that $y \in H^\times x H^\times$ (viz., $x$ and $y$ are associate). Moreover, we define, for every $x \in H$, $$ \mathscr{Z}_H(x) := \pi_H^{-1}(x) \cap \mathscr{F}^\ast(\mathcal A(H)) \subseteq \mathscr{F}^\ast(\mathcal A(H)) $$ (the set of factorizations of $x$) and $$\mathsf L_H(x) := \{\|\mathfrak a\|_H: \mathfrak a \in \mathscr{Z}_H(x)\}$$ (the set of lengths of $x$). We call $H$ atomic if $\mathsf L_H(x) \ne \emptyset$ for all $x \in H \setminus H^\times$.

Next, let $H$ and $K$ be multiplicatively written monoids, and let $\varphi$ be a homomorphism $H \to K$. We write $\varphi^\ast$ for the unique homomorphism $\mathscr{F}^\ast(H) \to \mathscr{F}^\ast(K)$ such that $\varphi^\ast(x) = \varphi(x)$ for all $x \in H$, and we say that $\varphi$ is essentially surjective if $K = K^\times \varphi(H) K^\times$ (this is actually an instance of the notion of essentially surjective functor in category theory), and an equimorphism (from $H$ to $K$) if the following hold:

1. $\varphi(x) = 1_K$ for some $x \in H$ only if $x \in H^\times$, that is, $\varphi^{-1}(1_K) \subseteq H^\times$.
2. $\varphi$ is atom-preserving, i.e., $\varphi(a) \in \mathcal A(K)$ for all $a \in \mathcal A(H)$.
3. If $x \in H \setminus \{1_H\}$ and $\mathfrak b \in \mathscr{Z}_K(\varphi(x)) \ne \emptyset$, then $(\mathfrak b, \varphi^\ast(\mathfrak a)) \in \mathscr{C}_K$ for some $\mathfrak{a} \in \mathscr{Z}_H(x)$.

Lastly, we say that $H$ is essentially equimorphic to $K$ if there exists an essentially surjective equimorphism from $H$ to $K$.